In general, you need to repeat the article. The conjunction and or or joins two noun groups, each of which need to include an article (which can be the null article sometimes, of course).
I have an orange and an apple in my hand.
I had some money three months ago, I could buy a television or a radio.
There is an exception for and, when you are not saying that there are two objects, but that there is a single object which can be designated by two nouns. For example:
I have a washing machine and a dryer. [I have two appliances.]
I have a washing machine and dryer. [I have a single appliance which has both functions.]
This is rare, however. In this example, the standard term would be “washer-dryer”. Without using that term, other possibilities include “washing machine and dryer combination” or “washing machine cum dryer” or “washing machine with dryer”.
It is possible to have a pattern like “a NOUN1 and NOUN2 NOUN3”, where “NOUN1 and NOUN2” is a complement of NOUN3. Here NOUN1 and NOUN2 take the null article; the article a is attached to NOUN3. So the article is repeated.
I have a steak and kidney pie in the oven.
I have a steak and kidney pie and an apple strudel in the oven.
With or, you can sometimes leave out the second article. There isn't a clear difference in meaning, but leaving out the article tends to indicate that there is a given object whose nature is in doubt, whereas including the article tends to indicate that there are two objects and there is a choice between these two. It is not common to leave out the second article.