As a one-liner:

perl -pe"BEGIN{<>} chomp if $.%2==0"
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Reads from STDIN or from the file provided as an argument.

>type data
Line 1: Header
Line 2: Keep me!
Line 3: Join me to the previous line.
Line 4: Keep me!
Line 5: Join me to the previous line.
Line 6: Keep me!
Line 7: Join me to the previous line.
Line 8: Keep me!
Line 9: Join me to the previous line.

>perl -pe"BEGIN{<>} chomp if $. % 2 == 0" data
Line 2: Keep me!Line 3: Join me to the previous line.
Line 4: Keep me!Line 5: Join me to the previous line.
Line 6: Keep me!Line 7: Join me to the previous line.
Line 8: Keep me!Line 9: Join me to the previous line.
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