I spend a lot of time at work so in order to allow myself to be a little more "productive", I use a LiveCD version of Linux Mint 14 on the workstations at work and connect to my domain at home (teamfate.org). Everything has worked fairly well for me up until tonight. I tried connecting and it just hung there for a minute and finally it said "Connection refused". I tried several more times to no avail. The website won't come up, nothing that I have running seems to be reachable remotely at this point.

Originally I thought that maybe the admins had seen several connections to said domain and might have blacklisted it, so I decided to see if I could use connectbot on my phone to connect. Again, no dice. Site wouldn't load on the phone either (Not using WiFi, using mobile network to access). Pings to the server show that there is connectivity so the network being down seems unlikely. When I left for work last night I didn't notice anything amiss -- everything seemed to be operating normally. I haven't made any significant changes since my last login except for installing the Steam beta. I suppose that could possibly be the issue, but again that's something that seems unlikely to me. The only other thing that I can remember that would be worth noting is that I did update software packages when I got home yesterday morning, and I remember one of them dealing with Webmin, but I would like to think that some issues would have presented themselves well before I left for work 12 hours later. Does anyone have any idea what could be going on? How likely is it that sshd, apache2, and vsftp would all suddenly stop?

If icmp (ping) is working, but you can't get in, that means the network stack is up, but that's all you can assume.
A WAG (wild A** guess) says the disk filled up, so eg ssh can't write temp files it needs etc. A full disk can bring everything down.

Another wild guess is that the update or whatever else you have installed recently has caused a Kernel Panic. A full disk isn't going to keep apache from responding.

Another potential option would be that you are on a dynamic address at home and your external IP changed. You might try using a dynamic dns service with a client. Even some routers support being the dynamic dns client.

what do the logs on the teamfate.org host say?