Hcfyk

$ join -j 2 -o 1.1,2.1 file file | awk '!seen[$1,$2]++ && !seen[$2,$1]++'
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e

This assumes that no line in the input file contains any whitespace. It also assumes that the file is sorted.

The join command creates the full cross product of the lines in the file. It does this by joining the file with itself on a non-existing field. The non-standard -j 2 may be replaced by -1 2 -2 2 (but not by -j2 unless you use GNU join).

The awk command reads the result of this and only outputs results that are pairs that has not yet been seen.

If you have ruby installed:

$ ruby -0777 -F'n' -lane '$F.combination(2) puts c.join(" ")' ip.txt
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e

• 
  -0777 slurp entire file (should be okay as it is mentioned in OP that file size is small)


• 
  -F'n' split based on newline, so each line will be an element in $F array


• 
  $F.combination(2) generate combinations 2 elements at a time


• 
  puts c.join(" ") print as required


• if input file can contain duplicates, use $F.uniq.combination(2)
  

for 3 elements at a time:

$ ruby -0777 -F'n' -lane '$F.combination(3) puts c.join(" ")' ip.txt
a b c
a b d
a b e
a c d
a c e
a d e
b c d
b c e
b d e
c d e

With perl (not generic)

$ perl -0777 -F'n' -lane 'for $i (0..$#F) 
 for $j ($i+1..$#F) 
 print "$F[$i] $F[$j]n" ' ip.txt
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e

With awk

$ awk ' a[NR]=$0 
 END for(i=1;i<=NR;i++)
 for(j=i+1;j<=NR;j++)
 print a[i], a[j] ' ip.txt 
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e

A python solution.
The input file is fed to itertools.combinations from the standard library, which generates 2-length tuples that are formatted and printed to standard output.

python3 -c 'from itertools import combinations
with open("file") as f:
 lines = (line.rstrip() for line in f)
 lines = (" ".format(x, y) for x, y in combinations(lines, 2))
 print(*lines, sep="n")
'

Here's one in pure shell.

test $# -gt 1 || exit
a=$1
shift
for f in "$@"
do
 echo $a $f
done
exec /bin/sh $0 "$@"

Example:

~ (137) $ sh test.sh $(cat file.dat)
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e
~ (138) $ 

Using Perl we can do it as shown:

$ perl -lne '
 push @A, $_} exit
a=$1
shift
for f in "$@"
do
 echo $a $f
done
exec /bin/sh $0 "$@"

Example:

~ (137) $ sh test.sh $(cat file.dat)
a b
a c
a d
a e
b c
b d
b e
c d
c e
d e
~ (138) $