Replacement backreferences (also called placeholders) are defined with \n syntax, not $n (this is perl-like backreference syntax).
Note you do not need groups here, though, since you want to wrap the whole lines with single quotation marks. This is also why you do not need the g flags, they are only necessary when you plan to find multiple matches on the same line, input string.
You can use the following POSIX BRE and ERE (the one with -E) solutions:
sed "s/..*/'&',/" file.txt > result.txt
sed -E "s/.+/'&',/" file.txt > result.txt
In the POSIX BRE (first) solution, ..* matches any char and then any 0 or more chars (thus emulating .+ common PCRE pattern). The POSIX ERE (second) solution uses .+ pattern to do the same. The & in the right-hand side is used to insert the whole match (aka \0). Certainly, you may enclose the whole match with capturing parentheses and then use \1, but that is redundant:
sed "s/\(..*\)/'\1',/" file.txt > result.txt
sed -E "s/(.+)/'\1',/" file.txt > result.txt
See the escaping, capturing parentheses in POSIX BRE must be escaped.
See the online sed demo.
s="Hello089";
sed "s/..*/'&',/" <<< "$s"
# => 'Hello089',
sed -E "s/.+/'&',/" <<< "$s"
# => 'Hello089',