The E4400 Series Service Guide is here:
http://literature.cdn.keysight.com/litweb/pdf/E4400-90335.pdfAccording to the Service Guide (page 5-24):
Calibration data will be preserved with the removal of the battery. However,
user data will be lost unless it is saved and later restored using an external
controller.
So, unless you had user waveforms, sequences, setups, etc. that you wanted on the unit, you should be fine.
Your firmware should also still be intact with a dead battery. Service software is available if you want to upgrade your firmware, or if you really need to reload the firmware for some reason.
Replace the battery, and go from there. With any luck there's not something else wrong with the unit.
Or, if you haven't messed with the unit yet and you purchased it as working (on ebay, "Condition: Used"), you might consider returning it. The RF section can be difficult to repair, and there's no schematics or detailed service information generally available. If the seller is facing a return, they will probably not have a problem if you want to at least try a new battery and still accept a return if the issue was not the battery. But I would ask and get this agreement before digging in.