Can you use "a" before "1/4" when there is no unit following?

Usually in formal writing, it's safest to actually write out the word you're using.

Any of the following are correct.

"a quarter of the people voted for..."
"one fourth of the people voted for..."
"a fourth of the people voted for..."

In your other example though "a 1/4 inch hole" could be correctly written as "a one-quarter inch hole". In this case "one-quarter" is an adjective describing a hole where the article belongs to your noun. However, if it's completely unavoidable you should presume people will read it as "one quarter" or "one fourth" rather than "quarter" or "fourth" and leave off the article.

In general English writing, you'd write out the "quarter," as you have already observed. However, there exist many technical uses in which general English writing style is inappropriate, as for example in plans prepared by architects.

And now you have to be careful, because architects are smart people who know that they transgress general English writing style, and who do not wish to appear ignorant for doing it. Therefore, architects have their developed their own, idiosyncratic conventions which suit architectural plans, and architects are quite sensitive to violations of these conventions. Architects ward off charges of ignorance by being idiosyncratic together, in a very specific way, as a profession.

So, you may be in trouble here. What a headache for you!

To write, "A 1/4 of the people voted for ...," is exceptionally awkward, but one is not to start a sentence with a numeral, so what can you do?

I always hate advice which tells you to avoid the problem by rephrasing, but in this case, rephrasing may be the best you can do.

However, if you will not rephrase, you may still have at least one way out. Can you make your "1/4" line up visually with other fractions, maybe in a bulleted list, somewhat as follows?

On election day,

• 1/4 voted for Johnson and only
• 1/5 for Pfufnik; but
• 1/3 chose Smith, who won.

Here, you are thinking like an accountant, taking advantage of the fact that most typefaces present all Arabic digits with identical widths. You still have a problem if this occurs, though:

On election day,

• 1/  4 voted for Johnson and only
• 1/10 for Pfufnik; but
• 1/  3 chose Smith, who won.

Good luck.

A and an represent the number 1 in speaking and writing as in- a girl, a boy, a pen, an orange, etc. So using them with 1/4 would be repeating the 1 used in the number, which is unnecessary. You can use it when you say a quarter as answered above.