Polysaccharides from non-cyclic sugars?
In theory and especially synthetically there is no real reason why one should restrict oneself to polysaccharides where all monomers are in cyclic forms. A linear monosaccharide could, for example, act as a bifurcating branching site with two different sugars attached to the aldehyde carbon to build up the required O,O-acetal. Another variant may be to have a methyl group block the second acetal oxygen. In any case, you want to avoid the aldehyde becoming a hemiacetal because these are too reactive and would degrade quickly.

$$\begin{align}\ce{\underset{aldehyde}{R-CH=O}} &amp;&amp; \ce{\underset{hemiacetal}{R-CH(OH)(OR)}} &amp;&amp; \ce{\underset{acetal}{R-CH(OR)(OR)}}\end{align}$$

You may want to look up acetal and acetalisation in the general organic chemistry textbook of your choice to learn more about why hemiacetals are much less stable than acetals.

In practice, however, linearised monomers in polysaccharides are not interesting. To the best of my knowledge, they do not occur in nature due to the labile hemiacetal intermediate that would be required — but to a much greater extent due to the fact that nature optimised the connection of cyclic monosaccharides into polysaccharides and is fully content with what it has. There is no need to evolve a synthase that is able to shield the highly reactive intermediates from the surroundings and thus it didn’t happen. While the formation of oligosaccharides can also be a spontaneous process given a catalyst, linear monomers in oligosaccharides will not form that way.

For most natural product or bioorganic chemists, it is by far more interesting to synthesise species that are known in nature or logical extensions thereof. While a linearised monosaccharide might qualify as an extension, it is not that logic an extension and thus of much lesser value.