Hi Squinky,
In few words, yes, you could run your fence off a battery that was being trickle charged from a solar panel but there are a few things to be aware of ( I'll try to explain it simply ).
If you look on the battery you're thinking of using, it will give you an Ah ( Amp hour ) figure. So if ( for example ) it says 140Ah, you could get 140 amps for one hour from it or, 1 amp for 140 hours.
Somewhere on your fence unit it will tell you the voltage ( 12v ) and wattage ( again, for example say 100w ).
Using the formula
P I x V
( Don't know how that comes out but it should read P divided by I times V )
Where P is the power in Watts ( w ), I is the current in amps, and V is the voltage.
We need to know I, so covering the unknown quantity gives us the formula to calculate it - in this case
P V
So,
100 12
Gives us an answer of 8.3A
If we use our battery above which has a capacity of 140Ah, we can calculate
140 8.3
Giving us a figure of 16.86 hours
I've set the wattage high so in normal use the battery wouldn't go flat in 16 hours but you can see how to do the calculations with your own unit.
You can use the same calculation for the output from your solar panel into the battery, but as Woodsman says, the output will be down slightly on the manufacturers quoted figures - I'd expect about 80% tbh, but maximise the output by orientating it so it faces south and isn't shadowed by anything.
A normal lead acid battery can be discharged to 20% of capacity and will recover fully when charged, a maintenance free gel cell battery can be discharged to 40% of capacity but once put on charge must be left until it is full.
I think you'd be asking a lot to run a pump off a battery, but you have the calculations now so you can do the maths and see if it will.
I hope that's helpful, and not too baffling!