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 fight you want to win.
The number is relative to the fight
You do not need many links in absolute terms. You need more than whoever is ahead of you in that specific search — and above all, more relevant ones.
For "plumber in Ashford", a few links from local directories and the trade association may be enough. For "life insurance", hundreds will not do it.
How to estimate it in practice
- Search for the term you want to win.
- Look at the first five results. Are they big, well-known sites, or sites about your size?
- If they are big, pick a more specific term. That decision is worth more than any number of links.
Quality changes the whole sum
Ten links from sites in your field are worth more than a thousand from random ones. 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 little by little.
Where to start
Instead of aiming at a number, make sure a few relevant links exist and keep appearing. A site that earns two good links a month is in better shape than one that bought three hundred and stopped.