#Perl, 16 +2 = 18 bytes (with regex)

Run with the -nl flags. -E is free.

say/^((.+)\2)+$/

Run as:

perl -nlE 'say/^((.+)\2)+$/'

Returns a list of the capture groups (a truthy) if pairable, null string if not pairable.

Explanation

The -nl flags will run the code in a loop (-n), putting the input (with trailing newline removed because of -l) into the variable $_ each time, then evaluating the code each time input is entered, until the program is manually terminated. The -E flag lets you evaluate code on the command line, and enables the say command.

say/^((.+)\2)+$/

   /^((.+)\2)+$/  #The regex engine
      (.+)\2      #Find any set of at least one character, followed by itself
     (      )+    #Do this at least one time
   /^         $/  #Make sure that this matches the entire string from start to end
say               #Output the result of the regex

If a match is found (e.g. if the string is pairable), then the regex returns a list of the capture groups, which evaluates to a truthy, which is then passed to say, and output. If no match is found, then the regex returns the empty string, which evaluates to falsy, which is then passed to say, and output.

Sample:

$ perl -nlE 'say/^((.+)\2)+$/'
aaababbabbabbbababbaabaabaababaaba
baababaababaaba                      #Truthy
baababbabaaaab
                                     #Falsy
bbababbb
bbb                                  #Truthy
aabaaababbbaba
bababa                               #Truthy
abaabaaba
                                     #Falsy

Perl, 16 +2 = 18 bytes (with regex)

Run with the -nl flags. -E is free.

say/^((.+)\2)+$/

Run as:

perl -nlE 'say/^((.+)\2)+$/'

Returns a list of the capture groups (a truthy) if pairable, null string if not pairable.

Explanation

The -nl flags will run the code in a loop (-n), putting the input (with trailing newline removed because of -l) into the variable $_ each time, then evaluating the code each time input is entered, until the program is manually terminated. The -E flag lets you evaluate code on the command line, and enables the say command.

say/^((.+)\2)+$/

   /^((.+)\2)+$/  #The regex engine
      (.+)\2      #Find any set of at least one character, followed by itself
     (      )+    #Do this at least one time
   /^         $/  #Make sure that this matches the entire string from start to end
say               #Output the result of the regex

If a match is found (e.g. if the string is pairable), then the regex returns a list of the capture groups, which evaluates to a truthy, which is then passed to say, and output. If no match is found, then the regex returns the empty string, which evaluates to falsy, which is then passed to say, and output.

Sample:

$ perl -nlE 'say/^((.+)\2)+$/'
aaababbabbabbbababbaabaabaababaaba
baababaababaaba                      #Truthy
baababbabaaaab
                                     #Falsy
bbababbb
bbb                                  #Truthy
aabaaababbbaba
bababa                               #Truthy
abaabaaba
                                     #Falsy