How do I carry out regression analysis with a sample size of only 28 and number of variables (including DV) 14?

For regression, this is way too less number of samples. It is advisable to use 20–30 samples par variable. Therefore, what you can do here, is consider the correlation of the independent variables with the dependent variables and choose the highest correlated variable to the dependent variable and build a one variable model once. You can also try out principal component analysis on the sample to create 3 effective variable to capture the variance mostly, but I suppose the effectiveness of the PCA is also not that good in a sample size of yours. I wonder whether your main task is to do regression or not. Because such small data are seen mostly in neuroscience and psychology where the main task is to find out underlying factors and not doing any prediction of sort. If you are also having similar reasons, then resort to tests like ANOVA, mANOVA, rank tests and others and devise them carefully enough to find out the effects you are trying to find out.

Finally, if you have a cost effective way to increase your data sample (remember, the data should ideally be independent of each other) then please do increase your data at least upto 500 or so. And then only you can see some good results.

For small samples, I found these links which you may find interesting to read:

(1) When small samples are problematic - Eiko Fried

(2) Problems with small sample sizes

Also, if you are interested in linear regression in general, please follow linear regression  in my blog.