Of course it should work with batteries. If you have checked that the batteries are OK and inserted correctly, and there isn't a switch, then the radio itself must be faulty. If its less than a year old, take it back and make a complaint.
All this is not to say that DAB radios will work for very long on batteries, as they tend to drain more current than an ordinary FM radio. I made the mistake a few years ago of buying a wind-up FM/DAB radio and no matter how long I wind the ruddy thing up, it won't work for very long on DAB, but is OK on FM !
You should have no problems accessing the instruction booklet on line, if it has been lost. What is the exact make and model please ?