A double-pole breaker with a "60" on each half will trip at 60 amps. The reason it's two connected breakers is to give the circuit 240V potential, by using two opposing AC phases. These do not "add up" to more current -- you're using them to double the voltage.
So if the heater requires 240V power at 120amps, you will need double-pole breakers that can provide a total of 120amps, e.g. 4 30A double-pole breakers as Tester101 suggested.
An electrician will be able to tell you if your service panel can accommodate this extra load -- it's a lot, but your 200A service might be adequate, depending on the normal load elsewhere in your home.
Put your computers on a UPS, even a small one. Just protecting yourself from short power outages will save you a bunch of downtime. Even if you resolve your overload issue, this is still worth it.
Most computer users today can get by with a laptop that's under $500, giving you built-in battery backup & portability in a compact, low-power package. Plug it in to your keyboard, mouse, and monitor the same as your desktop today, so your work experience doesn't change.
The power strip you linked to (http://www.cyberpowersystems.com/products/surge-protectors/home-surge/6050S.html?selectedTabId=specifications&imageI=#tab-box) doesn't appear to have a breaker. I think that whoever wrote that was just confused about what it means to be rated for 15A.
If your coffee maker and space heater are in the same location, you could plug them into mutually exclusive switched outlets. You'll need:
A steel square box, and an appropriate face plate
A regular duplex receptacle. 15A or 20A can work
a 3-way switch
cable
plug (15A or 20A, to match the recep)
fittings
Snap off the tab on the hot side of the recep, then run short leads from those 2 screws to the 3-way switch. This will let the switch choose one socket or the other. Plug in the heater on one and the coffee maker on the other. Now it's easy to make sure only one is in use at a time.
(Later I will add some pictures and other details. If anyone has pointers to the correct fittings and cable, please comment.)
Best Answer
I agree with others .33a no way for even the smallest ac units. At 5 minutes I believe the circuit is being overloaded.
Most breakers for residential use are inverse time breakers. A 15 amp breaker loaded at 15.5 amps may not trip for a 1/2 hour push it to 16 amps and it will trip much quicker possibly 5 minutes.
I would check the load on that breaker with a clamp meter and would expect more than 15 amps. If the load is less than 15 the breaker is defective and needs to be replaced.