I have data file:
0 0 3 -0.17 6 -0.05
0 0 3 -0.23 6 0.90
0 0 3 -0.41 6 0.50
0 0 3 -0.50 6 -0.33
0 0 3 -0.20 6 0.80
I want to plot the figure like this which connects each point in the lines. Can you tell me how?

I have data file:
0 0 3 -0.17 6 -0.05
0 0 3 -0.23 6 0.90
0 0 3 -0.41 6 0.50
0 0 3 -0.50 6 -0.33
0 0 3 -0.20 6 0.80
I want to plot the figure like this which connects each point in the lines. Can you tell me how?

The following suggestion uses the plotting styles with boxxyerror, with vectors and with labels.
The zero level gets the color of the last entry because everything is plotted on top of each other. Check the following example as starting point for further tweaking.
Code:
### energy diagram
reset session
$Data <<EOD
0 0 3 -0.17 6 -0.05 A_{one}
0 0 3 -0.23 6 0.90 B_{two}
0 0 3 -0.41 6 0.50 C_{three}
0 0 3 -0.50 6 -0.33 D_{four}
0 0 3 -0.20 6 0.80 E_{five}
EOD
set key noautotitle
set xrange [-1:8]
myWidth = 0.3
plot for [i=1:3] $Data u (column(2*i-1)):(column(i*2)):(myWidth):(0):0:xtic(i*2-1) w boxxy lw 3 lc var, \
for [i=1:2] '' u (column(2*i-1)+myWidth):(column(i*2)): \
(column(2*i+1)-column(2*i-1)-2*myWidth):(column(i*2+2)-column(i*2)):0 w vec nohead lw 1 dt 3 lc var, \
'' u 5:6:7:0 w labels offset 6,0 tc var font ",16"
### end of code
Result:
Addition:
From your comments: If the levels are to close such that the labels overlap, you could add an individual offset (here in column 8). Check the following example where the data is modified that the levels B and E are very close.
Code:
### energy diagram with individual offset of labels
reset session
$Data <<EOD
0 0 3 -0.17 6 -0.05 A_{one} 0
0 0 3 -0.23 6 0.90 B_{two} 0.05
0 0 3 -0.41 6 0.50 C_{three} 0
0 0 3 -0.50 6 -0.33 D_{four} 0
0 0 3 -0.20 6 0.85 E_{five} -0.05
EOD
set key noautotitle
set xrange [-1:8]
myWidth = 0.3
plot for [i=1:3] $Data u (column(2*i-1)):(column(i*2)):(myWidth):(0):0:xtic(i*2-1) w boxxy lw 3 lc var, \
for [i=1:2] '' u (column(2*i-1)+myWidth):(column(i*2)): \
(column(2*i+1)-column(2*i-1)-2*myWidth):(column(i*2+2)-column(i*2)):0 w vec nohead lw 1 dt 3 lc var, \
'' u 5:($6+$8):7:0 w labels offset 6,0 tc var font ",16"
### end of code
Result: