Translation:There is a small convenience store by the hospital.
You can think that the "no" particle in Japanese is something like "of" ("that belongs to") in English.
"Byouin no soba" => "near of a hospital" => "by the hospital". "Watashi no pen" => "pen of me" => "my pen".
"No" is posessive particle and restricts the right part to the left.
It works like all of the other directions, but there's not a good English analogue.
Just like you'd use の前に for in front of or 'at its front' (which is clunky but highlights the use of の), I like to think of this meaning 'at/in its "nearness"'. Nearness isn't really a word (as far as i know) but I think that's the concept.
There are two types of adjectives, い-adjectives and な-adjectives; some adjectives can be both, depending on the way they are written (for example 小さい vs 小さな). In this case, the adjective comes right before the name, with nothing in between, so it needs to be in the な form.
Grammatically, 小さなコンビニがあります means the konbini is the subject, marked with が, who does the verb あります by the hospital. In your sentence the hospital is the subject who exists by the konbini.
In other words, in your sentence it would mean you're explaining where the hospital is using the konbini as your point of reference, but the original sentence is the opposite.
The original feels like an answer to "where can I find a konbini?" and they say "there is one by the hospital", assuming the person who asked knows where the hospital is.