We have an older Toshiba DKT system in our workplace and the voicemail tends to quit working for awhile and then will start up again. Normally to access the mailbox we dial 250 but when this is done now we are hearing three beeps and then ringing at the other end. Any ideas on why this may be happening. Eventually it starts working again but it's usually not for a couple of days later.

If all the ports in the voicemail are busy it could say camp-on in your telephones display when you dial 250, or sound like your describing. If the voicemail isn't set up properly the ports can go into a constant loop back and won't release again until the voicemail does a clean up at 2:00 AM Tuesdays.

OK that makes sense. Any suggestions on how I may be able to fix this on my own or would I need a tech to come in and have a look at it?

I don't see any way you could fix this problem by yourself without calling a tech. There are so many things that could be happening here. There could be a problem with the voicemail ports getting locked up, or you may need more voicemail ports to support your call volume.