The answer to the question is "no" but the real issue is how to make do.
The furnace runs at about 100 watts.
Your batteries have about 12 usable watt hours per pound. (a typical RV battery weighs 50 - 70 pounds for about a kilowatt hour of electrical energy)
The fridge, alarms, and other such things run about 10 watts. (240 watt hours per day. maybe)
Many folks find that keeping the rig warm at night (65-75 degrees) with the furnace will do in a battery over night.
For battery state, use a digital voltmeter (see What’s the battery doing? Start with a DVM
). After a least a half hour of no significant charging or discharging, anything over 12.6v is a full battery. When it gets down to 12.4v you need to think about recharging, At 12.2v recharging is a high priority. The battery should not go below a resting voltage of 12.0v (assuming a 12v system).
The best thing you can do for your batteries is to make sure you have a converter that does 3 stage charging and has a maintenance mode that keeps a full charge on the battery when you aren't using it and implements a suflation inhibiting technique.
Learning how to live comfortably with the battery you have takes a bit of learning. Don't use the RV lights unless you really need to and then only when really needed. Use blankets to keep warm and the furnace just to keep from freezing except maybe to warm up things in the morning. We get so used to taking electrical power for granted it can be a bit of a challenge to avoid over-doing the battery.
If you really want to avoid running the battery down, start paranoid about using too much on batteries until you find out just what you can get away with.