What does "funding" mean for undergraduate students performing research over the holiday period?

There are two things that require funds for a summer studentship:

• Paying the student living costs
• The cost of the research materials themselves (which might be more or less important depending on the subject, but in my field we budget about £10,000 (~$15,000) a for a full time year per researcher.

As has been pointed out it depending on the local laws, you might legally count as an employee. I would expect that to be the case in many european countries. In fact its a big grey area in UK law, and working out if you an employee, a trainee or a volunteer is complicated.

But the law aside, I personally do not take unfunded summer students because the privilege those that have the means to support themselves without income for several months over those who can't. The system of unpaid internships in law, media, politics etc, has long been seen as a source of the unrepresentativeness of those professions and research is no different. There is a big move to here outlaw unpaid internships explicitly. I do not want to be part of that system and mine is not an uncommon position here.

You've been told by faculty members that they don't have the funds [to pay you]. Since you aren't interested in getting paid, you could approach faculty members again. Explain you're willing to work for free. Whether that's possible depends on the country's employment law.

Another answer discusses the ethics of working for free and argues that it benefits the privileged who can support themselves. Some good points are raised and I've upvoted that answer. I don't entirely agree though: Sure the privileged benefit, but the unprivileged do to. Only they do so at greater cost. Perhaps working two jobs, minimising expenditure, or racking up some debt. Ultimately, refusing unpaid interns deprives the unprivileged of a levelling-up opportunity.