Given a Diagram, Find X

Python 3, 428 408 385 378 bytes

Working, but has a ton of potential to be golfed. I'm a bit rusty.

Assumes input is padded with spaces to make a rectangle.

EDIT: Thanks to @Artyer for 23 byte savings!

EDIT2: Wow I completely missed a 6 byte savings. Saved 1 more by swapping sides of an equals check.

*i,=map(list,input().split('\n'))
r=c=s=q=e=w=0
o=lambda y,x:len(i[0])>x>=0<=y<len(i)
d='\/_|'
for l in i:
 if'x'in l:r=i.index(l);c=l.index('x')
for a,b in(1,0),(0,1),(-1,0),(0,-1):
 y,x=r+a,c+b;f=o(y,x)and i[y][x]
 if f in d:s=f;w=d.index(f);q,e=y,x
k=lambda y,x,g=[1,1,0,1][w],v=[1,-1,1,0][w]:o(y,x)and s==i[y][x]and(exec('i[y][x]=0')or 1+k(y+g,x+v)+k(y-g,x-v))
print(k(q,e))

Expanded version with explanation:

inputtt='''  ______________.
 / ____________ \
 |/ __________ \|
 ||/ ________ \||
 |||/ ______ \|||
 ||||/      \||||
 |||||/  x  |||||
 |||||\_____/||||
 ||||\_______/|||
 |||\_________/||
 ||\___________/|
  \_____________/'''

# First, we get the input from STDIN and make it
# into a doubly-nested array

*input_text,=map(list,inputtt.split('\n'))

# A pretty cool Python trick to assign 0 to
# mulitple variables at once.

row=col=line_letter=line_row=line_col=line_char_index=0

# A function to check if a certian row and col is
# in bounds or not. Uses python's comparator chaining

in_bounds=lambda y,x:len(input_text[0])>x>=0<=y<len(input_text)

# A string to store all the line characters.
chars='\/_|'

# Search for the x
for line in input_text:

 # If this line contains the x...
 if'x'in line:

     # Mark the row and column
     row=input_text.index(line);col=line.index('x')

# For each direction...
for down,right in(1,0),(0,1),(-1,0),(0,-1):

 # Move in that direction
 y,x=row+down,col+right

 # If the position is in bounds, mark the char at that position
 line_found=in_bounds(y,x)and input_text[y][x]

 # If the char is a line char, set all the variables saying we found it
 if line_found in chars:
  line_letter=line_found
  line_char_index=chars.index(line_found)
  line_row,line_col=y,x

recur=lambda y,x,\
       # Find which directions we are supposed to recur in based on the line char
       g=[1,1,0,1][line_char_index],v=[1,-1,1,0][line_char_index]:\
       # If the char is in bounds and we are still on the line...
       in_bounds(y,x)and input_text[y][x]==line_letter and\
       # Set the spot to a 0, so we won't go back, increment,
       # and recur in both directions
       (exec('i[y][x]=0')or 1+recur(y+g,x+v)+recur(y-g,x-v))

# Finally, print the answer
print(recur(line_row,line_col))

JavaScript (ES6), 165, 155 bytes

EDIT: Inlined x and w, to save some more bytes.

Golfed (assumes input is padded with spaces to make a rectangle)

t=>([k=e=o=1,v=t.search`\n`+2,-1,-v].some(h=>i=({"|":v-1,"_":1,"/":v,"\\":v})[d=t[r=l=t.search`x`+h]]),[...t].map(_=>o+=(k&=t[l-=i]==d)+(e&=t[r+=i]==d)),o)

Expanded

/*
   G(<input string,space padded>) => line length
*/
G=t=> {

    /*
     ! Note that these two are inlined, in the "golfed" version !
     "w" - line "width"
     "x" - index of "x"
    */
    x=t.search`x`;
    w=t.search`\n`+1;

    /*
    Locate the "line"    
     l,r - left cursor, right cursor (for navigating along the line)
     k - left stop flag, e - right stop flag
     i - increment
     d - direction (char)
    */
    [k=e=o=1,v=w+1,-1,-w-1].some(h=>i=({"|":w,"_":1,"/":v,"\\":v})[d=t[r=l=x+h]]);

    /* 
     Travel along the line axis in both directions
    
     Note, the stop condition should rather be: while(k|e), 
     but we iterate over all the chars with map instead (as o is guaranteed to be < # chars),
     to save some space
    */
    [...t].map(_=>o+=(k&=t[l-=i]==d)+(e&=t[r+=i]==d));

    /* 
      Resulting line length
    */
    return o;
};

Test

G= 
t=>([k=e=o=1,v=t.search`\n`+2,-1,-v].some(h=>i=({"|":v-1,"_":1,"/":v,"\\":v})[d=t[r=l=t.search`x`+h]]),[...t].map(_=>o+=(k&=t[l-=i]==d)+(e&=t[r+=i]==d)),o);

[
G( 
 "| \n" +
 "| \n" +
 "|x\n" +
 "| \n" +
 "| \n"
),

G(
"|\\   \n" +
"| \\x \n" +
"|  \\ \n" +
"|___\\\n"
),

G(
"Diagram of a Wobbly Line:\n" +
"IRRELEVANTTEXT____       \n" +
"____     ____\/           \n" +
"    \___\/ X ;)           \n" +    
"      x                  \n"
),

G(
" ______________ \n"  +
"/ ____________ \\\n" +
"|/ __________ \\|\n" +
"||/ ________ \\||\n" + 
"|||/ ______ \\|||\n" +
"||||/      \\||||\n" +
"|||||/  x  |||||\n"  +
"|||||\_____\/||||\n" +
"||||\_______\/|||\n" +
"|||\_________\/||\n" +
"||\___________\/|\n" +
" \_____________\/\n"
)
]

Sample output (if you run this in Google Chrome Developer Tools console)

[5, 4, 3, 5]