Learn French: Nous avons failli perdre les données.

Yes, faillir generally means 'to fail.' When you use it in the passé composé directly before an infinitive, you are literally saying you 'failed to do' that action by a very narrow margin. In English, we express this near-miss by saying we 'almost' did it.

Why use avoir failli + infinitive instead of just using presque?

Both are perfectly correct and mean the exact same thing. Nous avons presque perdu les données is very common. However, Nous avons failli perdre is a standard, idiomatic way to really emphasize the 'near miss' aspect, often highlighting that a negative outcome was narrowly avoided.

Can I use faillir in the present tense to say I am 'almost' doing something right now?

No. When used to mean 'almost,' faillir is strictly restricted to past tenses, most commonly the passé composé. If you want to say you are currently almost doing something, you must use presque.

Is the word for data (les données) always plural?

Usually, yes. Just like 'data' in English refers to a broad collection of information, it is overwhelmingly used in the plural in French. A single piece of information or a given fact is une donnée, but when talking about computer files or datasets, you will use les données.