Does the SVM require lots of features most of the time?
I stumbled upon some rules of thumb for dataset sizes, but not specific to the ratio of features / samples.

Also, it seems easier to guess to the "positive" side, i.e. "might it work?" then "might it fail?". 
I witnessed both small datasets (&lt;1000) that functioned well with even 20-50 features, and huge datasets (~10M) that functioned well with very few (&lt;5) features.

The stage in the processing pipeline to tackle this, is the feature selection one. I'd start with gathering all of your make-sense features and playing with different k-best values for the feature selection object to find. (ranges like: 10, 50, 100, ..)

You didn't mention a specific research environment, but I'd like to suggest scikit-learn sklearn.feature_selection.SelectKBest running inside a GridSearchCV pipeline. This way the grid object will help you choose the closest k-best featues out of a given list. Something like:

from sklearn.feature_selection import SelectKBest
from sklearn.pipeline import make_pipeline
from sklearn.model_selection import GridSearchCV
from sklearn.svm import LinearSVC

pipe = []
pipe.append(SelectKBest())
pipe.append(LinearSVC())

est_pipe = make_pipeline(*pipe)
print 'all possible grid params: {0}'.format(est_pipe.get_params().keys())
param_grid = {
  # some svc grid params..,
  'selectkbest__k': [5, 10, 50, 100]
}

grid = GridSearchCV(est_pipe, param_grid=param_grid)
grid.fit(X, y)