Yes you can stain and paint it but as sidecar-jon said it's not a good plan. It will just look like brown paint. If it's plain mdf really it needs painting to look any good.
now keeping in mind any stain or coloured varnish can only work on what base colour you have
what you can do which wont look very good but better than plain mdf is
to take coloured varnish[wicks light oak accrylic quick drying for example]
brush it on a bit heavy and overuse the brush a bit till you get a grain effect
i have done this several times the effect is poor[in my oppinion] but without using paint dont think youll get grain effect
as said above youl get coloured nothingness
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