I am compiling the following assembly program on debian wheezy, but it will not run giving me the error:

-bash: ./power: cannot execute binary file

Code

.section data

.section text

.global _start
    _start:
        # do first calc and save the answer
        pushl $3
        pushl $2
        call power
        addl $8, %esp
        pushl %eax

        # do second calc
        pushl $2
        pushl $5
        call power
        addl $8, %esp

        # add both together
        popl %ebx
        addl %eax, %ebx

        # exit with answer as return status
        movl $1, %eax
        int $0x80

.type power,  @function
    power:
        # ?
        pushl %ebp
        movl %esp, %ebp
        subl $4, %esp

        # load params
        movl 8(%ebp), %ebx
        movl 12(%ebp), %ecx
        movl %ebx, -4(%ebp)

    power_loop_start:
        # have we looped down to 1?
        cmpl $1, %ecx
        je end_power

        # multiply prev result by base and store
        movl -4(%ebp), %eax
        imull %ebx, %eax
        movl %eax, -4(%ebp)

        # go again
        decl %ecx
        jmp power_loop_start

    end_power:
        movl -4(%ebp), %eax
        movl %ebp, %esp
        popl %ebp
        ret

I run with:

as power.s -o power.o
ld power.o -o power
./power

Both uname -m and arch give me i686, and the binary outputs this on objdump -x:

$ objdump -x power

power:     file format elf32-i386
power
architecture: i386, flags 0x00000012:
EXEC_P, HAS_SYMS
start address 0x00000000

Sections:
Idx Name          Size      VMA       LMA       File off  Algn
  0 text          0000004a  00000000  00000000  00000034  2**0
                  CONTENTS, READONLY
SYMBOL TABLE:
00000000 l    d  text   00000000 text
00000023 l     F text   00000000 power
00000032 l       text   00000000 power_loop_start
00000043 l       text   00000000 end_power
00000000 g       text   00000000 _start
08049034 g       *ABS*  00000000 __bss_start
08049034 g       *ABS*  00000000 _edata
08049034 g       *ABS*  00000000 _end

Not sure what I am doing wrong.

Further note :

This example is from the book "Programming from the ground up". I have tried on a Red Hat x86_64 machine, with as flags --32 and ld flags -m elf_i386, it all compiles fine just like on the x86 machine, but when executing gives the same error.

You have a typo: .section text should be .section .text (note the dot) or just .text

While the issue is caused by a typo I feel it is worthy of some explanation especially since you so nicely provided all the details :)

While you can name your sections however you like (and this is why the tools didn't complain), each section has some flags. In this case you see in the objdump output: CONTENTS, READONLY. What this means is that this section is not executable, in fact it's not even loaded. (Arguably the error message could be a little more precise.)

Okay, so why isn't it executable? The assembler recognizes some common section names and sets up the flags properly. For a custom name, you have to do that manually, for example by doing .section text, "ax" which sets ALLOC and CODE. See also the .section directive in the manual and this answer about the various flags.