Learn Hardcore Finnish: Avain on otettava mukaan, muuten en pääse sisään. - The key must be taken along, otherwise I can’t get inside.

Because Avain on otettava is a passive-like obligation construction: literally “The key is to be taken.”
In Finnish passives, the “object” often surfaces as a nominative “subject-like” noun.

Compare:

Active (you/I take it): (Minä) otan avaimen mukaan. = I take the key with me.
Obligation with an explicit person: Minun on otettava avain/avaimen mukaan. = I have to take the key with me.
(Both avain and avaimen occur in real usage here; avaimen is the more “object-like” form, while avain is common especially when the sentence feels more passive/impersonal.)
Impersonal/passive obligation: Avain on otettava mukaan. = The key must be taken along.

So avain is nominative because the sentence is phrased like “the key must be taken,” not “someone must take the key.”

What exactly is on otettava grammatically?

It’s olla (to be) + the necessitative participle otettava (from ottaa, to take).
This structure expresses necessity/obligation:

on tehtävä = must be done / has to be done
on mentävä = must go / have to leave
on otettava = must take

It’s similar in meaning to täytyy / pitää, but it often sounds a bit more formal, written, or “instruction-like.”

Who is supposed to take the key? The sentence doesn’t say I/you/we.

That’s intentional: this construction can be agentless (impersonal). It focuses on the requirement itself: the key must be taken.

If you want to name the responsible person, Finnish often uses a genitive + on + otettava:

Minun on otettava avain mukaan. = I have to take the key with me.
Sinun on otettava avain mukaan. = You have to take the key with you.

Why is mukaan used here? Does it mean “with”?

mukaan is an adverb/postposition meaning “along (with), with you, taking with” in this common pattern:

ottaa mukaan = to take along / to bring with you
tarvita mukaan = to need to bring along

So Avain on otettava mukaan means “The key must be taken along (with you/us).”

Could I replace this with Avain pitää/täytyy ottaa mukaan? Is there a difference?

Yes, those are very natural alternatives:

Avain pitää ottaa mukaan.
Avain täytyy ottaa mukaan.

Meaning is essentially the same. Differences are mostly about style:

on otettava: a bit more formal/instructional, common in notices, rules, written Finnish
pitää/täytyy ottaa: very common in everyday speech

What does muuten mean here, and why is there a comma?

muuten means “otherwise / if not”. The comma separates two clauses:

requirement: Avain on otettava mukaan,
consequence: muuten en pääse sisään.

So it’s the pattern: Do X, otherwise Y.

How does en pääse work? Why is en a separate word?

Finnish negation uses a special negative verb that carries person/number:

en = I don’t
et = you don’t
ei = he/she/it doesn’t
emme / ette / eivät = we/you(pl)/they don’t

The main verb then appears in a special form (often called the connegative):

pääsen = I get / I can get (in)
en pääse = I don’t get / I can’t get (in)

What does päästä mean here? Is it “can” or “get”?

päästä often means to get (to/into/out of) or to gain access, and it can imply ability/opportunity in context.

So en pääse sisään is naturally:

I can’t get in.
I’m not able to get inside / I won’t be able to enter (without it).

It’s about access rather than permission.

Why sisään and not sisälle? Aren’t both “inside”?

Both relate to “inside,” but they differ slightly:

sisään is a common adverb meaning “in / indoors / inside (to the inside)”, especially with verbs of entering:
mennä sisään, päästä sisään = go/get in
sisälle (allative of sisä-) can feel a bit more spatial/explicit, like “to the inside (of something),” and is also correct in many contexts:
mennä sisälle = go inside

In everyday Finnish, pääse sisään is extremely common for “get in (through a door).”

Why is present tense used (on otettava, en pääse) if it’s about the future (“if I don’t take it, I won’t be able to”)?

Finnish often uses the present tense for near-future or general consequences, especially in conditional/otherwise-type statements. The time is understood from context:

Ota avain mukaan, muuten en pääse sisään.
Literally present, but understood as: Take the key, otherwise I won’t be able to get in (later).

If you want to make the “future” more explicit, you can add time words (e.g. sitten, huomenna) or use other phrasing, but the present is the normal choice here.