It's really hard to say. The problem is that what might be considered a strong correlation varies across disciplines.
For example: I ask two people to measure the height of a group of children and correlate them. If I find a correlation of 0.7 I'm going to consider that very weak, and think that something has gone wrong somewhere.
Similarly, if I give those same children two different tests of cognitive ability, and find a correlation of 0.7, that would seem to be about the ballpark I'd expect.
If I try to correlate that cognitive ability with some sort of measure of behavior, if I found a correlation of 0.7 I'll assume I've put the wrong variables in my model. If I get 0.3 I'll be pretty excited.
In the social sciences Jacob Cohen suggested a rule of thumb which has become codified into some sort of law (and is often misunderstood).
- A correlation of 0.1 is small.
- A correlation of 0.3 is medium.
- A correlation of 0.5 is large.
But these are very general guidelines that are often (in my opinion) overinterpreted and/or misunderstood. (I've seen submitted articles where the authors have written [something like] "The correlation was 0.48 so it was medium, as it was below 0.5". The wikipedia page has more: https://en.wikipedia.org/wiki/Effect_size
Some interesting correlations are also presented on this page: http://web.sonoma.edu/users/s/smithh/psysurvey490/toc/effectsize.pdf For example, the correlation between ever having smoked and having lung cancer is around 0.1. A small correlation, but given the seriousness of cancer, you might not want to say that.