Technically, no, you can't store more than a byte value to a char type. In C, a char and a byte are the same size, but not necessarily limited to 8 bits. Many standards bodies tend to use the term "octet" for an exactly-8-bit value.
If you look inside limits.h (from memory), you'll see the CHAR_BIT symbol (among others) telling you how many bits are actually used for a char and, if this is large enough then, yes, it can store the value 1000.
Technically, no, you can't store more than a byte value to a char type. In C, a char and a byte are the same size, but not necessarily limited to 8 bits. Many standards bodies tend to use the term "octet" for an exactly-8-bit value.
If you look inside limits.h (from memory), you'll see the CHAR_BIT symbol (among others) telling you how many bits are actually used for a char and, if this is large enough then, yes, it can store the value 1000.
The range of values you can store in a C type depends on its size, and that is not specified by C, it depends on the architecture. A char type has a minimum of 8 bits. And typically (almost universally) that's also its maximum (you can check it in your limits.h).
Hence, in a char you will be able to store from -128 to 127, or from 0 to 255 (signed or unsigned).
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That is because of the way two's complement encoding works: 0 is treated as a "positive" number (signed bit off), so, therefore, the number of available positive values is reduced by one.
In ones' complement encoding (which is not very common nowadays, but in the olden days, it was), there were separate values for +0 and -0, and so the range for an 8-bit quantity is -127 to +127.
In 8-bit 2's complement encoding numbers -128 and +128 have the same representation: 10000000. So, the designer of the hardware is presented with an obvious dilemma: how to interpret bit-pattern 10000000. Formally, it will work either way. If they decide to interpret it as +128, the resultant range will be -127..+128. If they decide to interpret it as -128, the resultant range will be -128..+127.
In actual real-life 2's complement representation the latter approach is chosen because it satisfies the following nice convention: all bit-patterns with 1 in higher-order bit represent negative numbers.
It is worth noting though, that language specification does not require 2's-complement implementations to treat the 100...0 bit pattern as a valid value in any signed integer type. E.g. implementations are allowed to restrict 8-bit signed char to -127..+127 range and regard 10000000 as an invalid bit combination (trap representation).