One approach would be to reset the input record separator.

#!/usr/bin/perl

use warnings;
use strict;

$/ = "\n> ";

while (<DATA>) {
    chomp;
    print;
}

__DATA__
.06669701$-.1672469$.02157899$.0346167$.65879324$.91614802$.45012441$r
+s11080516
-.05563282$-.00141932$.00036263$.00035752$.00541792$.00345471$.9269254
+3$rs11080
> 530
-.00042649$-.00475721$.00167316$.00182299$.99057815$.90393977$.6990673
+3$rs11080
> 537
.10901125$.0361255$.02148475$.00853908$.74113541$.45908988$.68361003$r
+s11080542
-.03866776$-.05004879$.00185491$.00145606$.38131545$.36141448$.1755140
+3$rs11080
> 557
-.004521$.01312692$.00033174$.00070174$.77873394$.88615378$.63499741$r
+s1108056
.0339248$.02398934$.00276444$.00303152$.80478053$.55026576$.67512938$r
+s11080561
[download]

outputs:

.06669701$-.1672469$.02157899$.0346167$.65879324$.91614802$.45012441$r
+s11080516
-.05563282$-.00141932$.00036263$.00035752$.00541792$.00345471$.9269254
+3$rs11080530
-.00042649$-.00475721$.00167316$.00182299$.99057815$.90393977$.6990673
+3$rs11080537
.10901125$.0361255$.02148475$.00853908$.74113541$.45908988$.68361003$r
+s11080542
-.03866776$-.05004879$.00185491$.00145606$.38131545$.36141448$.1755140
+3$rs11080557
-.004521$.01312692$.00033174$.00070174$.77873394$.88615378$.63499741$r
+s1108056
.0339248$.02398934$.00276444$.00303152$.80478053$.55026576$.67512938$r
+s11080561
[download]

EDIT:
The simple example I gave just outputs the data, but, instead, you could push each line onto an array or append it to a string for further processing.