As a file/print server, a netbook would work fine. Only problem is limited expansion. I just went through the process of researching and setting up a server from older hardware, and I ended up going with a big case with a ton of bays instead of using external drives with the small enclosure it was in. I'm glad I have, because I've already got 5 HDDs in it and more will come... I wouldn't be able to use that many externals, especially not with a netbook with 2 or 3 USB ports.
However, the 3Ghz Pentium-4/512MB DDR-533 in my server is plenty fast on Arch Linux (which is, to be fair, MUCH lighter on resources than Windows or even Ubuntu Server), and can serve Samba shares, take on image backups, stream video and music, and act as a DNS server all at the same time without slowing down or stuttering. I just end up running out of network bandwidth first. So a netbook should be fine for the uses you're planning on giving it.
As for the always-on concern, as long as you don't stuff it into a tiny cabinet, you'll be fine. Just run it with the battery removed unless you want to use it on the go. That will help it run cooler. Of course, you can also point a box fan at it too... that will make the battery removal moot and let you use the battery as a UPS.
I do LOVE having a home server though. I can't believe it took me this long to finally get one up. I can access anything I want anywhere in the house, and if a hard drive dies, fixing it is as simple as loading the old disk image onto the replacement. Additionally, using the server for DNS makes general browsing noticeably snappier.