It's a little bit of both, in my opinion - but both tasks are DIY'able.
First you need to find where the leak is - turn the taps on as you would normally, look under the sink and see where any water comes from - as above, if drips aren't immediately visible, wrapping kitchen towel around the pipe joints can give you a clue.
Secondly, the old flex outlet could do with being checked. If it is badly corroded, an electrician may want to isolate it.
If it is ok, it could be covered with a blanking plate, and a little silicone around the edges wouldn't hurt.