'some degree of' - a little or a lot?

When "some degree of" is used, it's generally a case of deliberate ambiguity. The speaker is talking about a value that is not easily quantifiable, either because the property is not measurable or because the variables are sufficiently 'fuzzy' that picking a single value is too difficult.

So in the case of your car example, the writer/speaker knows that they can do the repair but (at the time of the statement) they aren't in a position to know exactly how much work that would involve. If they commit to a particular amount (say 8 hours) and then discover during the work that the repair is more complicated, the car owner might might not be happy to wait/pay for the extra time. Keeping the estimate ambiguous avoids creating an unnecessary expectation.