In theory, anything is possible if you have enough money to spend on it! Here are some thoughts to consider:

1. The wall you want to move may be structural, in which case you will need to add a beam to support the structure above. In general, if the wall is perpendicular to your joists, it is a supporting wall, but don't be fooled if it runs parallel to the joists - there may still be something funky going on inside. You can remove sheetrock to take a look without requiring a permit, BUT do not remove anything steel or wood without checking with a structural engineer first. If there is nothing structural, electrical or plumbing in this wall, you can remove it without a permit.
2. Consider that there may also be services in this wall - electrical wires, water pipes, HVAC ducts or gas pipes, all of which would need to be moved by a professional. The further they need to be moved out of the way, the more it is going to cost you.
3. The width of the opening will determine the size of the beam - the wider the opening, the deeper the beam will be. This will determine how much the beam soffit 'drops' into the space. If you want a truly flush ceiling, this will involve cutting every joist in order to push the beam up and hang every joist from it. More work = more $$$.
4. If the span is very wide, you may need to spend more on a steel beam, or consider a mid span column. Steel can also be considered if you want to keep the beam smaller than a wood one would allow.
5. Don't forget about finishing the floor. Hopefully the floors either side of the wall you just removed are at the same level and the same material, and you will be able to infill the gap with matching finish. If not, you need to decide what finish this is going to be and any transition height you may need to have. Anything over half an inch is not going to be ideal.