Is it technically possible to build a ship that could keep a constant 1G acceleration / deceleration on a flight from Earth to Mars? What could provide energy? What could provide accelerant? (I am not asking if it would be affordable, though a guess at the cost would be fun to see )
There are simply two options: either you provide 1 g of acceleration on the spacecraft the entire flight (from thrust) or you use a rotating habitat to provide centrifugal acceleration. Applying 1 g of acceleration is probably not possible given the fact your trajectory will have to be planned to accommodate that (i.e., you will be expending a ridiculous amount of propellant to achieve this), so a rotating habitat would be the most feasible option.
In that case, the energy required is minimal -- either just enough to overcome friction between the rotating and non-rotating components of the spacecraft, or essentially zero if the entire spacecraft itself rotates.
But... Let's consider some ballpark estimates if you still insist on trying to achieve a 1 g continuous acceleration from thrust...
If you want to do this with chemical rocket motors, you'll be looking at a manoeuvre that needs so much fuel my calculator simply says "error" (basically you'll need a $\Delta v$ of over 2500 km/s, assuming a 3 day flight).
If you want to use ion thrusters then you'll still need to achieve the same $\Delta v$ but your specific impulse will be much higher (say 6,000 s instead of 350 s). This leads to a mass ratio of about 6E18, or rather 6 billion billion (so you will need over 6,000,000,000,000,000,000 kg of propellant for every 1 kg of empty mass). Not to mention the assumptions made here in terms of how you will power this and provide the thrust on that huge mass to get 1 g of acceleration.
Note that I'm approaching this question in terms of accessible technology. You could look into what kind of specific thrust would make this achievable for a more acceptable mass ratio, or some kind of external source of propulsion, but at that point you are making up propulsion technology that does not exist!