With the expected advances in reusable rocket technology within the next decade, will constructing a space elevator become meaningless? Or will it still be way cheaper, safer and convenient to use them?
A Space Elevator would still be amazingly useful
The two factors that come to mind are forms of power and scale:
With a space elevator connected to the ground, you could use the energy in your power grid to lift everything up. By doing this, we can use green and renewable energies.
With rockets, baring any massive advancements, we are restricted to using chemical energy. So not only to we have to specifically manufacture the fuel, but the entire process maybe less efficient. Not to mention the environmental effects of burning millions of tons of methane if you want something big in orbit.
Most likely, we'd be able to bring much larger masses into orbit at a time with a space elevator. Even with SpaceX's BFR system, a large ship in orbit has to refuel multiple time to get anywhere. With a space elevator, we could potentially bring a fully fueled ship (without any need to have in-atmosphere propulsion on-board) into orbit in one go.
All this being said, we still need HUGE advances in material science for any space elevator to be viable.
Rockets suffer from the tyrrany of the rocket equation. While a reusable rocket is great, you need to burn a lot of propellant for each kg of mass to reach orbital velocity.
A space elevator can steal momentum from the planet itself to provide the speed required to reach orbital velocity (or beyond).
In general, fast energy expenditure is less efficient than slow. Elevators allow efficient energy (solar power or whatever) to be used to lift cargo (electric climbing cars). Rockets require high energy density fuel (rocket fuel) to be burned and lifted to the spot where it is used by earlier rocket fuel.
The 'hard' part of a space evevator is the materials science required; we lack matter that has the tensile strength required on human scales in a controlled lab, let alone industrial scales in the open air. Once that is solved, we have the problem of building it in place, building a factory in orbit, dodging debris and weather, designing lifters, etc, each of which are problems larger in scale than the Apollo Program.
Reusable rockets would reduce the cost of lifting stuff to orbit, but the fuel costs would remain high. Such cheap launch technology would open up the possibility of building (expensive) factories in space, and/or move asteroids into position to provide the other end of the space elevator. So quite possibly they are a prerequisite for the elevator.
Once you have the elevator, lifting to orbit energy cost plummets and it becomes basically electricity (at current prices of 1$ per kg to orbital velocity). Currently it costs on the order of 10000\$ per kg. Reliable reusable rockets might cut that down, but probably to 1000\$ or 100\$s of dollars per kg, not single-dollars per kg.
Space elevators have limited capacity, so while the energy costs might be 1\$ per kg, prices may be similar to competing technology until demand saturates the pipe. However, the first space elevator can then be used to build the second, and you'd have something similar to an industrial revolution exponential economic expansion as the cost of reaching space converges on the energy cost.
At 100\$-1000\$/kg to orbit, space travel of humans requires a small fortune per person. Space may open up to industry and robots, but not to migration.
At 1\$/kg to orbit, and 2\$/kg to Lithobraking on Mars (not including travel time), the solar system becomes our back yard.
Whatever concept of space elevator you want to build, it requires you to transport massive amounts of material into space, so no. Having cheap launch vehicles is actually a requirement to building a space elevator. Whether the elevator could ever break even given its high mass is another matter altogether.
A space elevator, no matter what sort of unobtanium it's built from or how it works, has one unbeatable advantage over a rocket: the elevator cars don't need to carry their fuel with them. The vast majority of the energy involved in a rocket launch goes into carrying rocket fuel to ever-higher speeds and altitudes; being able to leave your energy source on the ground is a huge efficiency gain.
The phrase tyranny of the rocket equation isn't just poetry.
Another interesting point - the hard part isn't getting up, it's going fast.
Presumably, since the elevator is tethered to the ground it's already going fast - fast enough to stay in geostationary uh... orbit?
Since by the time whatever you were moving into space got to the top of the elevator, it's going to be going the same speed, it should be a lot easier since you just need to keep moving.
In the book "The Fountains of Paradise", by Arthur C. Clarke, it is mentioned that a space elevator can regenerate some of the electricity it uses to move material up by electronically braking as it brings material down, like regenerative braking on electric cars.
There is also the possibility of actual rescue, should something go wrong. There is a very intense section of the book that involves a rescue maneuver, which also involves "thinking outside the box". I'd explain more, but I don't want to drop any spoilers.
I believe the book also mentions that with the space elevator, you don't need to reach escape velocity until you reach the end of the elevator. Since that's in space, escape velocity from that point should be considerably less than it is from the surface. Less air resistance should also make it easier to reach.
Rockets work on a exponential thrust restriction. The more you want to put into space, the way more fuel you need to lift off. It's not just that you are lifting 10 lbs into orbit, but you are also lifting the all fuel to get you there. The rocket crews have to make decisions of how important items are to a mission, because that 10 lbs of cargo may mean something like 10,000 lbs more fuel. Maybe this is explained by the tyranny of the rocket equation, but I don't have the chops to understand that.
Because there's so much fuel going into a rocket launch, it's expensive. Electricity, on the other hand, is relatively cheap. Renewable and nuclear generators have pretty much guaranteed that. Plus, with the extra height of the space elevator to get past the atmosphere, solar panels might be able to be added to the tower itself, allowing it to generate some of it's own power, beyond the regenerative braking mentioned earlier.
Beyond launch things into space, a space elevator could be a starting point for a ring around the earth, at orbital height. This could in turn be a platform for a full planet enclosure. Both ideas are referenced in several books I've read but can't recall right now. Both have their advantages in economy and beyond.
Going back to "The Fountains of Paradise", it tells about families going for sight seeing trips, including elderly people. Sightseeing is very big business, and if even the baby and the older generation of a family can join in, then you've really got something that'll draw visitors. Well, you know, besides being able to see a significant portion of the earth all at once, being "in space", and lots of other "cool factors."
The book also mentions being able to launch capsules in very short order, like multiples a day. Right now, that's not possible with rockets, even with airplane launched rockets.
The book is basically an essay in how space elevators are pretty much necessary for us to continue to be in space, couched in a science fiction novel. It's a great book, winning both the Hugo and Nebula awards. I've enjoyed reading it multiple times.
Escape velocity is greater than geosynchronous velocity. The asteroid or whatever at the end of a tether would have to travel a bit faster than geosynchronous velocity, because of all the mass below the orbit of the asteroid moving at slower than (for the altitude of each bit) orbital velocity. Perhaps this would be close, or at least close enough, to escape velocity to allow easy exit from Earth's gravity well.
But even so, the tether itself would have to be very strong to resist the tension it would have to bear. A lot of people have suggested that carbon nanotubes would do the trick; I suspect even that would not be enough.
And even if we could invent the materials, how would we construct such a thing? If we capture an asteroid, put it into low-Earth orbit, and move the materials to it, we would still have to make a connection somehow. We can't just lower the cable into the atmosphere -- it would be whipping around until we (somehow) grab onto the end and attach it onto the Earth station.
Perhaps we will solve all these problems someday. But it's a little more difficult than I believe most people think.