Bill,
It rather depends on how you do it:
1) You could have a dedicated server; this costs lots of money, but is ideal if you have a very popular or very big website.
2) You could have a "virtual server" - which looks just like a real one to me, but is actually a very clever program running on another machine; so you could have 20 or 30 "pretend" servers running on one real one, if it were powerful enough.
3) You could have space on a shared server - regular web hosting. The disadvantage to this is having sufficient control to be able to run a complex forum program AND database, etc.
I plumped for a virtual server last time listerengine was down (I bought it the day before it came back online!) It's the cheapest one I could find, but it's easily powerful enough to run this forum - which is actually quite a small one, as these things go. It costs me £10 (about $15) per month; and I get unlimited bandwidth with that.
If I had to hazard a guess, I'd say each message consumes, on average, 4KB (it's probably less than this; 1KB of pure text is said to be around a sheet of A4 (or Letter) of single-spaced typing; but there are inefficiencies in the storage). So 50K messages would use 200,000KB; or 20Mb [roughly]. I have 9.1Gb of disk space available, so I could fit the forums in 400-times over with space to spare.
I know it'll run the forum software, because iit already is. So: if I had a backup of the forums as they are now, I could have it all up & running in a couple of hours. The problem is, getting that backup: So far, that's not been possible...
The reason I know all this stuff? I've lived & breathed computers for 22 years (since I was 13 years of age)... the irony is that I'm now thoroughly enjoying tinkering with a super-low-tech diesel engine which is nearly 25 years older than me!