This is basically the suggestion from Fabian Claremont's answer, but (for beginners) put into code and visualized. Actually, Ciro Santilli's solution is even shorter.
Tested with gnuplot 5.2. For gnuplot 4.6 (the time of OP's question), replace reset session with reset and set margin 8,-1,1-1 with set lmargin 8 and set bmargin 1.
Code:
### Multiplot with single y-label
reset session
unset key
set sample 500
set multiplot layout 7,1
    unset ylabel
    set linetype 1 lc rgb "red"
    set margins 8,-1,1,-1    # left, right, bottom, top (-1=auto)
    set ytic 1.0
    set title "Plot 1" offset 0,-1
    plot sin(1*x)
    set title "Plot 2" offset 0,-1
    plot sin(2*x)
    set title "Plot 3" offset 0,-1
    plot sin(3*x)
    set title "Plot 4" offset 0,-1
    plot sin(4*x)
    set title "Plot 5" offset 0,-1
    plot sin(5*x)
    set title "Plot 6" offset 0,-1
    plot sin(6*x)
    set title "Plot 7" offset 0,-1
    plot sin(7*x)
    set lmargin -1   # automatic lmargin
    unset title
    set origin 0,0
    set size 1,1
    set border 0
    unset tics
    set ylabel "This is a centered y-label"
    plot [][0:1] -1     # plot a dummy line out of range
unset multiplot
### end of code
Result:
