What are the minimal sample size requirements for cross-validation or bootstrapping? I hope it makes sense to even ask these questions, but I'm wondering how can I evaluate the "validation" procedures that my data allow me to perform (i.e. cross-validation or bootstrap: nested/double or not, with blocks or not, stratified or not, etc.)?
I might initially think that I have a fairly large sample size ($n$) compared to the number of my predictors ($p$) but if I have to create blocks or strata to account for peculiar data structures, surely the ratio $n/p$ quickly decreases.

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*Are there rules-of-thumb regarding the minimal sample size required to perform cross-validation (e.g. for each fold of a k-fold CV)?

*What about bootstrapping?

*Does it depend on my modelling goals? What questions should I ask myself to help me choose the appropriate "validation" strategy?

I'm new to all this so please forgive me if I mix things up or use inappropriate terms.

The more difficult sample size estimation problems are (1) $n$ needed for reliable model development and (2) $n$ needed for reliable external validation.  For internal validation using resampling, averaging 100 repeats of 10-fold cross-validation will seldom provide an unreliable estimate of likely future performance, and the same can be said for running 400-600 bootstrap resamples, unless $n < p$ where the bootstrap will underestimate the amount of overfitting (but still usually tell you the modeling process is awful).
Said another way, it is not too difficult to estimate how poorly a model is likely to perform in future data from the same series.  It can be quite difficult to get the model to perform well.
The key to resampling is repeating all supervised learning steps, and sometimes also repeating unsupervised learning steps.
