My first sawmill rig was a beam machine on a 46cc Poulan ... not really usable, even on small diameter logs.
The second rig was an Alaskan Small Log mill using a MS390 (64cc) with a 20" bar and a Granberg Pico ripping chain. I used this to mill several cherry and maple logs - the largest about 16" diameter. This is certainly doable, but it takes time. Roughly 10-15 min (start to finish) to cut a cherry board 12" wide, 10' long. This is enough saw to be usable, but not very efficient - plan to spend an entire afternoon to mill one log. I would say this is practical enough to make a few boards and have some fun - it was enough to get me hooked on chainsaw milling!
My latest rig is an Alaskan 36" mill using an MS650 (85cc) with a 32" bar and a Woodland ripping chain. This rig takes about a third of the time to mill the same log (but it is much heavier). Some others have a different opinion, but I think the ripping chain makes a huge difference. My next step (hopefully) is to build a Procut-type rig.
I got the MS650 on eBay for a pretty good price, although you are taking a chance buying a chainsaw off the web - but I think I got lucky. I also got the Alaskan 36" mill on ebay, for less than $100, but I had to add some missing parts.