A lot depends on how much shooting you want to do and what calibers.
I built a bullet trap for my basement. It is about the size of a file cabinet. Most of it is made of wood, except for the backstop which is 3/8" steel at a 45 degree angle. This directs the bullets downward into a sandbox lined with 1/8" steel. The sandbox is about 12" deep and the sand itself is about 8" deep. It does not require a great deal of metalworking to build one like this - the biggest issue is getting the steel cut to size and a lot of suppliers will do that part for you. Then it is essentially a matter of building a big wooden box.
Ventilation - it depends. I tend to just shoot a few five shot groups at at time - maybe 25 rounds at most - and .22 caliber only. The smoke/vapor/air pollution is minimal. A couple of bathroom type fans exhausting out a basement window is plenty sufficient for this. Naturally, if I was going to have a big shoot-fest with a lot of people all blazing away with .45 Colts and black powder revolvers, I would need a much more serious ventilation system.
Hanging some old blankets around the basement helps deaden the sound a lot. This is more for the shooter than for neighborhood disturbance issues - people standing outside the house don't seem to hear or notice at all, even in my yard standing right next to the house. My wife will often be out working in the garden or whatever and come in later with no idea I've been shooting.
I said .22's only, but I know my bullet trap is adequate for more than that. If I am working on a new hand load or whatever, I will ocassionally shoot off a few test rounds in a light target load. I do not make a regular habit of shooting centerfire in the basement, though, because it is heaps more smoky/lead-y/etc. - ventilation and noise are the limiting factor, not the bullet trap.
All in all it is a lot of fun and well worth the effort. Trying to get out to the range can be a difficult proposition if it is a busy week, but I can always find 15-20 minutes to loose off a few .22's in the basement. Also, I find that shooting a lot of .22 improves my shooting much moreso than shooting a lot of centerfire.