Draw my bar graph

sh + awk + tac, 173

Mostly an awk script that prints the graph bottom up which is then reversed by tac.

awk '{n[NR]=$NF;$NF="";$0=p" + "$0;p=" |"p}1;END{for(print p;p~/\|/;print p (k>($0=0)?"|":""))for(i=k=p="";i<NR;p=p (j>0||k>0?"|":" ")(!k||$0?"_":" ")){j=k;k=n[++i]--}}'|tac

Description

awk, first part, executed for each input line

{
  n[NR]=$NF;         # store the value, here n[1]=0, n[2]=6, n[3]=3, n[4]=5
  $NF="";            # remove the value from the label
  $0=p" + "$0;       # add a prefix (initially empty) and a " + " in the front
  p=" |"p            # grow the prefix for the next string
}1;                  # implicitly print $0

Output

 + Bars can be 0 
 | + Large-bars_are$nice2 
 | | + average)(@#$ 
 | | | + neato 

awk, second part, executed once in the end

END{
  for(print p;p~/\|/;print p (k>($0=0)?"|":""))
    for(i=k=p="";i<NR;p=p (j>0||k>0?"|":" ")(!k||$0?"_":" "))
      {j=k;k=n[++i]--}}

ungolfed:

END{
  print p;           # print the prefix again
  for(;p~/\|/;)      # for each line, bottom up. no more "|" -> we're done
  {
    p="";            # string to be built
    i=k=0;           # i: bar index, k: heigth of the previous bar
    for(;i<NR;)      # for each bar, left to right
    {
      j=k;           # store the previous bars heigth in j
      k=n[++i]--;    # get the current bars remaining height and decrement it
      p=p (j>0||k>0?"|":" ")(!k||$0?"_":" ");
                     # if the bar to the left or this one have some height
                       remaining, draw a border in between them, else a space
                     # if this bars remaining heigth is exactly 0, draw a top
                     # if $0 is set, draw the bottom
    }
    print p (k>0?"|":"");
                     # draw (or not) the rightmost border, finally print
    $0=0;            # unset $0, only to detect first run
  }
}

Output

 | | | |             # the prefix
 _|_|_|_|            # the strings built by the nested loops
  | | | |            |
  | | | |            v
  | |_| |
  | | | |
  | |  _
   _                 # no more "|" in the string, we're done

tac reverses the lines

   _    
  | |  _
  | | | |
  | |_| |
  | | | |
  | | | |
 _|_|_|_|
 | | | |
 | | | + neato 
 | | + average)(@#$ 
 | + Large-bars_are$nice2 
 + Bars can be 0 

JavaScript (ES6), 270 262 270 287

Bug fix added a missing row of '|' under the bars

l=>{t=p=o='';l=l.split`
`.map(r=>([,b,c]=r.match(/(.*) (\d+)/),[' + '+b,+c>t?t=c:c]));for(++t;t--;o+=`
`)l.map(x=>o+=x[1]<t?'y y':x[1]>t?t?'x x':'x_x':'y_y');return o.replace(/(yx)|(xy)|(xx?)/g,'|').replace(/y+/g,' ')+[...l,' '].map(x=>p+x[p+=' |',0]).reverse().join`
`}

Test Test in Firefox, as Chrome does not support ES6 Destructuring assignment

F=l=>{
  t=p=o='';l=l.split`\n`.map(r=>([,b,c]=r.match(/(.*) (\d+)/),[' + '+b,+c>t?t=c:c]));
  for(++t;t--;o+=`\n`)l.map(x=>o+=x[1]<t?'y y':x[1]>t?t?'x x':'x_x':'y_y');
  return o.replace(/(yx)|(xy)|(xx?)/g,'|').replace(/y+/g,' ')
  +[...l,' '].map(x=>p+x[p+=' |',0]).reverse().join`\n`
}

function test()
{
  var i=I.value
  O.textContent=F(i)
}  
  
test()

textarea { display:block; width:50%; height:5em}

Input
<textarea id=I>Bars can be 0 0
Large-bars_are$nice2 6
average)(@#$ 3
neato 5</textarea>
<button onclick='test()'>go</button><br>
Output
<pre id=O></pre>

421 bytes - Python 2

import sys
l=sys.stdin.read().split('\n')
b=[(' '.join(x[:-1]),int(x[-1])) for x in map(str.split,l[:-1])]
a=len(b)
e=enumerate
m=[' '*(a+1)+'|'*x[1] for i,x in e(b)]+[' '*(len(b)+1)+'|'*b[-1][1]]
h=[' '*i+'+'+'|'*(a-i)+'_'+' '*(x[1]-1)+'_' for i,x in e(b)]
c=m+h
c[::2]=m
c[1::2]=h
c=[''.join(' ' if not x else x for x in l) for l in map(None,*c)]
for i,(n,_) in e(b):
 c[a-i-1]+='\b'*i*2+n
c='\n'.join(c[::-1])
print(c)

Tests

a 1
b 2
c 3
     _
   _| |
 _| | |
|_|_|_|
 | | |
 | | + c
 | + b
 + a