Is it possible for a website owner to steal a user's password and somehow gain access to their computer or files?
In general.....no. But if you have downloaded anything from his site, he could have included a trojan in that download.
Now he will have access to the password you use to access his site and be able to know what OS you are using and other various pieces of information that your computer broadcasts to the world.

There have been some new exploits where computer have been comprimised by visiting a web site, but those exploits have not been around long.
Get a² Free : http://www.emsisoft.com/en/software/free/ and run a scan with that. Also get AdAware : http://www.lavasoftusa.com/software/adaware/ and Spybot S&D : http://www.safer-networking.org/en/index.html and run those also. I would run an outside AV scan such as Trend 'Housecall' also: http://housecall.trendmicro.com/hous...start_corp.asp

Personally, I would suggest you also change OSes also. Either XP Home or XP Pro and upgrade it immeadiately to SP2, but that is a personal preference. Although either XP with SP2 is more secure than Windows ME.

Good luck

edit> all programs listed above are free, BTW.
If you do upgrade to XP, then do a complete reformat and then installation and then you will be sure you have gotten any malware off of your computer (you can also reformate and reinstall ME to do the same thing).