When will I have a binary car?

Javascript, 68 63 61 60 52 bytes

5 bytes off thanks @ETHproductions. 2 3 11!! bytes off thanks @NotthatCharles.

f=(i,a,m=0)=>/[^01]/.test(i+=a[m++%10])?f(i,a,m):m/2

Test here.

MATL, 29 26 25 bytes

`tvyyYs+V50<!A~]xx1Mf1)2/

Input format is

[27; 27; 27; 27; 27; 27; 27; 27; 27; 27]
101101

EDIT (June 10, 2016): The following link replaces v by &v (26 bytes) to adapt to changes in the language

Try it online!

`              ]           % do---while loop. Exit loop when top of stack is falsy
 t                         % duplicate. Takes first input (array) first time
  v                        % concat vertically (doubles length of array)
   yy                      % copy top two. Takes second input (todasy's value) first time
     Ys                    % cumulative sum of (repeated) miles each way
       +                   % add to today's value
        V                  % convert each number to a string
         50<!A             % true for strings than only contain '01 ' (ASCII less than 50)
              ~            % negate. This is the loop condition
                xx         % delete stack contents
                  1M       % push again array that tells which strings only contain '01 '
                    f1)    % pick index of first true entry
                       2/  % divide by 2

Lua, 108 Bytes

First time using the repeat..until loop in a codegolf!

function f(o,t)i=0repeat i,o=i+1,(o+t[i%#t+1]).."."o=o:sub(1,o:find("%.")-1)until tonumber(o,2)print(i/2)end

Ungolfed

function f(o,t)               -- o can either be a number or a string, t is a table
                              -- call it like f(2,{27,14,5...})
  i=0                         -- initialise the travel count
  repeat                      -- proceed like a do..while in C
    i,o=i+1,(o+t[i%#t+1]).."."-- increment i, and add t[i] to the odometer
    o=o:sub(1,o:find("%.")-1) -- remove the decimal part of the odometer
  until tonumber(o,2)         -- tries to convert o from binary to decimal
                              -- return nil (and repeat the loop) if it can't
  print(i/2)                  -- print i/2 which is the number of days for i travels
end

After the first loop, o will have a decimal part because of tonumber, I had to remove it... And to add it for the first case, that's why I concatenate it with a ".".