#Java 10, 339 308 265 227 225 186 184 bytes
Java 10, 339 308 265 227 225 186 184 bytes
v->{var s="v->{var s=%c%s%1$c;return 0+new java.math.BigInteger(s.format(s,34,s).getBytes()).toString(2);}";return 0+new java.math.BigInteger(s.format(s,34,s).getBytes()).toString(2);}
-8 bytes thanks to @NahuelFouilleul removing the unnecessary &255 (and an additional -35 for bringing to my attention that the full program specs of the challenge had been revoked and a function is allowed now as well..)
-41 bytes thanks to @OlivierGrégoire.
Explanation:
quine-part:
var scontains the unformatted source code String%sis used to put this String into itself withs.format(...)%c,%1$cand34are used to format the double-quotes (")s.format(s,34,s)puts it all together
Challenge part:
v->{ // Method with empty unused parameter and String return-type
var s="..."; // Unformatted source code String
return 0+ // Return, with a leading "0":
new java.math.BigInteger( // A BigInteger of:
s.format(s,34,s) // The actual source code String
.getBytes()) // Converted to a list of bytes (UTF-8 by default)
.toString(2);} // And convert this BigInteger to a binary-String