A very big one --- to give you a short answer.
To avoid wasting a lot of money, you should consult the Owner's Manual that came with the inverter or better still, call the agents or the manufacturer.
If I understand your question correctly, you want to run a 5,000 Watt inverter from a 12 Volt battery when the main supply to the inverter fails. If this is the case, the following will apply.
Since there would be little point in having a 5,000 Watt inverter unless you wanted to connect a load that is close to 5,000 Watts, the battery is going to have to be able to provide 5,000 Watts to the inverter. Were the DC input to the inverter, 12 Volts, the battery would have to be able to supply the inverter with 417 Amperes. A bank of three Numax 12 Volt 135 Amp-Hour batteries, connected in parallel (starting & deep-cycle) would be able to keep the inverting running for one hour but each one would be supplying 135 Amperes which would soon damage the batteries. To keep the current down to 45 Amperes, you would have to connect nine of these batteries in parallel (if you want an output of 5,000 Watts to be sustainable for one hour, without damaging the batteries.
I cannot and do not claim to be an expert on batteries but I suspect that even 45 Amperes is a lot of current to draw from a 135 A-H battery, for more than a few minutes. The adviser at the battery shop will be better able to advise you on how much current can be drawn for a long period.