How can you combine all lines that end with a backslash character?

a shorter and simpler sed solution:

sed  '
: again
/\\$/ {
    N
    s/\\\n//
    t again
}
' textfile

or one-liner if using GNU sed:

sed ':x; /\\$/ { N; s/\\\n//; tx }' textfile

Here's an awk solution. If a line ends with a \, strip the backslash and print the line with no terminating newline; otherwise print the line with a terminating newline.

awk '{if (sub(/\\$/,"")) printf "%s", $0; else print $0}'

It's also not too bad in sed, though awk is obviously more readable.

It is possibly easiest with perl (since perl is like sed and awk, I hope it is acceptable to you):

perl -p -e 's/\\\n//'