Output the Arecibo Message

05AB1E, 215 210 200 bytes

Saved 15 bytes thanks to Magic Octopus Urn

•cOž¤4é57ñΛ\Ö₃BαöĀíL½₅üBdoÙRθLγ¨G×Tćú$G(˜ƒ¦!€R»SDrµCnJ†d∊ζ·<8‡T@|‹ï=BζćósxG\ÙÎ$¿o₁5/ÔŸÇBûXé-”a::Ž]°∊y;ζ]MÜβ‘иL”β{üÃÇíäc€÷›ÎU=}¨иaŸdY`»¾ÚUβ:ô©¦β†₅DGŠβ3Jêθ,äá!ícqšVÖ›lÈΣ¯pε €êʃDpÙ/¬Žλ8:ãÿ3=€.Þć•3BY¾4×:

Try it online! or with Additional formatting

Base-255 encoded trinary string with occurrences of 0000 replaced by 2.

Java, 688 678 590 379 361 bytes

Returns a string.

n->new java.math.BigInteger("in95mzupnpa2r0khpoepyql6ioqyn413avucdtfay6indx4wh9dehe3sn18klobtf4z9g9q17umqmwpegr2khb5eqinn7azl4jpfp2a8eui0xfrx5qwrou6gd65jh4ge3ls14k5lu7qrvmg6942ms29u5rb8fa6yrdhfoh5zoi9bdi7uh5ig0u0ff9kounth8sh357x7qox4m3oqviqsbrvakonbka4ahp21bgzi5v1akzzuqoncszhpabbru9q1uo2g11zr73iuyiqr5ikr69zn7cdv7e1lhd6ese9",36).toString(3).replace("2","0000")

-10 bytes by returning the raw stream (old answer)
-88 bytes by using base 10 numerics (thanks @ceilingcat!)
-211 bytes (I knew it could be golfed!) by using a base-36 encoded BigInteger (thanks @JollyJoker!)
-18 bytes by using a different encoded integer (thanks again @JollyJoker)

Try it online!

Explanation:

n->new java.math.BigInteger("base36 string",36) // Decode the base-36 integer.
   .toString(3)                                 // Re-encode as ternary
   .replace("2","0000")                         // Replace 2 with "0000"
                                                // Implicit return

Jelly, 213 bytes

“H²ɓ¶Ṡḷ€ẹ]ƒf*ḳḢ&ƁṇOḥ{ḄṫwỊ+oLạʋߢH9¢¹÷ỴɗÇ⁶ƲƙæḊẋ3³=1!VƇƁ'D⁺3Ỵɱ©⁵%fȯez#ƈjƒżṆo.ZF⁶ċṢ⁶ọṛb9Ȯƒd?ƁUĠt4ẇ,ḞġƒµƭfʠƁP§÷øȤŻPɲẋ(¢ß¢(⁽3¶ṙėɗy@ṁYȮL~e⁷ƤĊ§nỊṅµṠ°@7ẠB>Ġ⁻İ}uy¡½:esOpḢt}qS©HÞṬĖṛṇḣ9÷;ESḢ,Ẉ^ṙpƲ©tṃwçnẒṆ¡⁻Jıƒị£-&Ɱ*ẋʂżoȯÑḢɼ’

Try it online!

I played around with Huffman coding, but the improvements in data size were outweighed by the extra code. As such, this is simply a base-250 encoded version of the desired output. Output consists of an integer that when decoded as bijective base 2 will yield the 1D list of 1s and 2s. Thanks @Emigna for pointing out the change in rules.

Try it online - with further decoding to demonstrate output!

If a more conventional binary encoding is preferred, here is one that encodes an integer representation of the inverted binary message. The most significant bit of the integer represents the beginning of the message.