Linux

awk '
/^>/ {                                  # If current record starts with > ( /^</ )
        $0 = (NR > 1 ? RS $0 : $0)      # If current record number is greater 1 (NR > 1) set it to newline followed by current record (RS $0)
        ORS = RS                        # Set Output Record Separator to Record Separator (ORS = RS) [ RS is newline by default ]
}

! />/ {                                 # If current record does not contains pattern > ( !/>/ )
        ORS = ""                        # Set Output Record Separator to "" (ORS = "")
}

END {                                   # END Block
        printf "\n"                     # Print newline
} 1                                     # 1 == true, so print current record
' file

$ sed -n "/^>/{x;s/\n//2gp;}; /^>/! {H;}; $ {x;s/\n//2gp;};" file
>some lines of text
actgtgaaactgtgacgtcg
>some lines of text
acgtgcagtcgtttgcgt