The question has a boring answer and a useful one. The boring one is "it depends". The useful one is: look at who is winning the contest you want to win.
The number is relative to the contest
You do not need many links in absolute terms. You need more than whoever is ahead of you for that specific search — and, above all, more relevant ones.
For "plumber in Tacoma", a few links from local directories and the trade association may be enough. For "life insurance", not even hundreds will do it.
How to estimate in practice
- Search Google for the term you want to win.
- Look at the first five results. Are they large, well-known sites, or sites the size of yours?
- If they are large, choose a more specific term. That decision is worth more than any quantity of links.
Quality changes the whole calculation
Ten links from sites in your field are worth more than a thousand from random sites. That is not a figure of speech: irrelevant links tend to be simply ignored, so a thousand of them can be worth literally zero.
Pace matters
A new site that gains five hundred links in a week draws attention for the wrong reason. Steady growth looks like what it is — a site being discovered gradually.
Where to start
Instead of aiming at a number, make sure a few relevant links exist and that they keep appearing over time. Publishing your listing across a network of search sites in your field, like Magneth, covers the base — and how to get quality backlinks covers the rest.