Learn Hardcore Russian: В клинике можно получать медицинскую помощь бесплатно. - In the clinic, one can receive medical care for free.

Клиника is a feminine noun (dictionary form, nominative singular).
After the preposition в with the meaning “in/at (location)”, Russian usually uses the prepositional case.
The prepositional singular of клиника is клинике.

So в клинике = “in the clinic / at the clinic” in the sense of location.

Why is there no subject like “you” or “people” in this sentence?

Russian often uses impersonal constructions to express general possibility or necessity.

Можно
- infinitive means “it is possible to … / one can … / you can …”.
There is no explicit subject; the sentence describes a general situation.

So В клинике можно получать… literally is “In the clinic it is possible to receive…”, which in natural English is “You can receive…” or “One can receive…”, without specifying exactly who.

What exactly does можно do here? Is it a verb?

Можно is an impersonal modal word meaning “it is possible / may / can”.

It is used with an infinitive: можно получать = “(one) can receive / it is possible to receive”.
Grammatically, it behaves a bit like an impersonal predicate (there is no subject: no “I/you/he” attached to it).

So можно gives the meaning of possibility/permission rather than describing an action by a particular person.

Why is it получать and not получить? What is the difference here?

The difference is aspect:

Получать – imperfective: focuses on the process or repeated action (“to receive, to be receiving, to get (in general)”).
Получить – perfective: focuses on the result / one completed act (“to receive (once), to get (successfully)”).

With можно, both are possible but the nuance changes:

В клинике можно получать медицинскую помощь бесплатно.
Emphasizes that in general, you can receive medical help there, as an ongoing or repeated possibility.
В клинике можно получить медицинскую помощь бесплатно.
Emphasizes that it is possible to get medical help there (at least once / as a concrete result).

In this sentence, получать fits well because the statement is about the general, ongoing possibility of getting free medical care at that clinic.

Why is it медицинскую помощь, not медицинская помощь?

Медицинская помощь is the nominative form (“medical help” as the subject of a sentence).

Here, медицинскую помощь is the direct object of the action “to receive”, so it must be in the accusative case:

Feminine noun помощь (help) – accusative singular is the same as nominative: помощь.
Adjective медицинский must agree with помощь:
- Feminine nominative: медицинская помощь
- Feminine accusative: медицинскую помощь

So:

получать (что?) медицинскую помощь – “to receive (what?) medical help.”

Why is помощь feminine, and how do I know that?

Помощь is grammatically feminine. Many feminine nouns end in -ь (soft sign) and are irregular-looking.

You can tell it’s feminine here because:

The adjective with it is медицинскую, which is feminine accusative singular.
Adjectives must agree in gender, number, and case with the noun.

So the form медицинскую помощь tells you помощь is feminine.

Why is it бесплатно and not бесплатная or бесплатная помощь?

Бесплатно is an adverb meaning “for free”.

In Russian, it is very common to use an adverb in this kind of structure:

получать что-то бесплатно – “to receive something for free”.

Alternatives:

бесплатная медицинская помощь – “free medical help” (here бесплатная is an adjective modifying помощь).
получать бесплатную медицинскую помощь – “to receive free medical help.”

Your sentence chooses to say “receive medical help for free” rather than “receive free medical help”, but the meaning is similar. The grammar is just different: adverb (бесплатно) vs adjective (бесплатная/бесплатную).

Can the word order be different? For example, can I say Медицинскую помощь можно получать в клинике бесплатно?

Yes, Russian word order is relatively flexible, and your sentence is grammatically correct:

Медицинскую помощь можно получать в клинике бесплатно.

The basic informational structure changes slightly:

В клинике можно получать медицинскую помощь бесплатно.
Starts with “In the clinic…”, focusing first on the location.
Медицинскую помощь можно получать в клинике бесплатно.
Starts with “Medical help…”, emphasizing what can be received.

Both are normal; context and emphasis decide which is more natural.

Could I leave out получать and just say В клинике можно медицинскую помощь бесплатно?

No, that would be ungrammatical.

Можно needs an infinitive verb to express what is possible:
можно (что делать?) получать – “it is possible (to do what?) to receive”.

If you remove получать, you just have:

В клинике можно медицинскую помощь бесплатно – there is no verb telling what can be done with the medical help.

You must keep получать (or another infinitive, like получить, получать, получать там etc.) for the sentence to be complete.

What is the difference between в клинике and words like в больнице or в поликлинике?

All three are locations related to healthcare, but they are not identical:

Клиника – often suggests a clinic (sometimes private or specialized), not necessarily a huge hospital.
Больница – hospital, usually where patients stay overnight, with inpatient departments.
Поликлиника – an outpatient clinic, where you go for appointments, tests, procedures, but you don’t stay overnight.

In this sentence, в клинике is neutral: “at the clinic”. In real usage, the choice of word can give extra information about what kind of medical institution it is.