You only need the core rulebook
While there are two Alien Archives now, you only need a single one of them to have a bunch of alien statblocks to work as antagonists. Yes, they also work as NPCs if you want thugs, but most will work best as monster encounters for the players to defeat. There are some new technological items in the books as well, usually related to the alien presented there.
But all the content from both Alien Archives should be available for free on online resources, such as the unofficial SRD, or the official Archives of Nethys. Note that their description is also available (Example: Mi-Go).
On the other hand, using online resources, you could build your own aliens, using the provided guidelines for Creating New Creatures and NPCs, which provide you with everything you need to create your own encounters. Everything else is up to your creativity and how you want those mechanics to work. You want an alien that can petrify with their gaze, or have a fetid aura, or paralyze those bitten. It's all possible, just check the Universal Monster Rules or take inspiration from existing aliens.
It is also possible to convert all monsters from Pathfinder 1e to Starfinder, using the Conversion Guidelines, giving you six full bestiaries and a few additional monster books to work with. This is the route I went with, and converting them is an easy task (for me, at least), granting me far more content than what has been published so far.
I don't have a copy of the Armory book, so I can't help there. I know it expanded the equipment list, such as adding far more Weapons, Armors, Augmentations, Fusions, Upgrades and rules for different Manufacturers, and I also know it does not fix one of the biggest complaints about the core rulebook, being able to build/modify a starship using credits. But other than that, I can't say much about it.
Best Answer
You can get the content of the first 8 books or so for free, legally, from WotC's site:
Originally found here: http://www.wizards.com/default.asp?x=d20/article/srd35
As of today (April 5th, 2019) while this link is still active, the files 404, but the Internet Archive still has them. (See links in comments) The rtf files contain the majority of the content of the following:
This is referred to as the SRD, or System Reference Document, and is legally free content released by Wizards of the Coast.
It has more than what you need to play the game. It contains all the rules text, but is missing the fluff text.
It is formatted differently than the books, being organized by subject instead of by chapter. Some people find this easier, others find it harder to find things.
There are also online repositories of this content, for example the http://www.d20srd.org/index.htm is highly regarded and often referenced.
Most people consider the 3 "core" books to be the ideal minimum to have, as the DMG has rules for environments, advancement, traps, dungeons, cities, and treasures; and the Monster Manual had a list of ready made monsters, which are both quite handy, if not strictly needed.
It is also entirely possible to play with only the 3rd edition Player's Handbook, and referencing 5th ed material. Some conversion on the fly will be necessary, and it will be challenging and require more work, but it can be done.
After all, this is a game of imagination, and Gygax himself once commented (and I paraphrase) that if everyone figured out that they could just make up their own rules, they would be out of a job. You don't actually need any books to play, just a set of rules that everyone can agree to. 7