There is a great deal of misinformation in the answers here.
To be pedantic, there is no possible way to remain stationary and at rest with respect to the sun. Everything with mass in the universe is exerting a force on you, so truely stationary is never possible.
The next best you can get, is at the core of the sun. If stationary there, you will remain stationary, but this is not a stable state, and the temps and pressures make remaining there quite difficult.
The next best to that is at a very large disance away. The further you go, the less influence the suns gravity will have on your position. If you are ignoring the small influences of everything else in the universe, you would remain stationary wrt. the sun only when the suns influence is ignored bybeing so far away that it also gets ignored. This is now a matter of how accurately you want to measure the problem,or in other words, how much motion you want to ignore. The motion is still technically there.
Within our solar system, and within relatively human timescales, temperatures and pressures, there is no passive way to remain stationary with respect to the sun. Your heliocentric motion will become apparent and measurable.
Of course, if you have fuel and thrusters to overcome this motion, then remaining stationary wrt the sun is possible, as long as your fuel reserves don't run out and your engine remains operational. There is no passive situation where you can come to rest wrt the sun and simply stay like that.