There weren't any to begin with of course! All the early rules mention flag posts, used to determine the boundary of the pitch. Whenever I watch children playing, they don't seem to take much notice of pitch boundaries, and it's fairly safe to assume that early players were the same! There were rules for throwing or kicking the ball back into play, so a brave umpire might agree the ball was "out" if one of the players appealed. As the football ground was often a cricket ground or a public park, it would not have been an easy matter to draw lines on the grass anyway!
Once a club had acquired its own ground, it became somewhat easier to mark out boundary lines without upsetting other users. Kick-offs required the centre of the ground to be marked. A rule that players had to be 10 yards away from a kick-off meant that a circle 10 yards in radius was drawn around the centre spot. We can assume that the four boundaries would have been painted. We also know that some bright groundsman, tired of having to repaint the lines, decided to dig a small groove in the ground instead. This idea was later banned by the authorities!
In 1887, an FA Committee issued notes of guidance to umpires and referees. These notes suggested a need for a line running right across the field to divide it into two halves, a semi-circle of one yard radius at the corner flags, and semi-circles of 6 yards radius round each goal post.
The arrival of the penalty in 1891 required two lines to be marked across the field, 12 and 18 yards from each goal line. (The penalty kick was originally given for offences 12 yards from the goal line. The kick could be taken from any position on the 12 yard line; the other players had to stand behind the ball and at least 6 yards from it. In the 1901 Cup final photograph below, the 12 yard line is quite clearly marked, the 18 yard line less so).
These arrangements lasted until 1902, when the markings we know today were introduced. The penalty kick law now applied to offences 18 yards from the goal line, but within an 44 yard-wide area known as the "penalty area". The penalty spot was now fixed, 12 yards from the goal and in the centre. A "goal area" 20 yards wide by 6 yards deep was introduced to replace the two semi-circles. There was one last little change to make, in 1937; originally players could stand anywhere along the 18 yard line while a penalty was taken; a section of a circle with a 10 yard radius was added to give the penalty taker a little more room for his run up.
Incidentally, you non-Brits and non-Americans reading this, it is interesting that we still have a goal mouth 8 feet by 8 yards. I know there is a metric equivalent, though not so easy to remember perhaps! The English rugby code was quick to embrace the change to metric measurements; the 25 yard line soon became the 22 metre line. Strange that soccer has not made the change!
The Tottenham Hotspur v. Sheffield United FA Cup Final of 1901, with the old pitch markings. Click on the picture to see a larger version.
The Newcastle United v. Aston Villa FA Cup Final of 1905, with the new markings. Click on the picture to see a larger version.