Factors affecting the intestinal-barrier permeability of newborns, such as body mass index (BMI), nutrition and antibiotics, are assumed to affect intestinal-barrier permeability in the first two years of life. This study assessed 100 healthy, full-term newborns to 24 months old. Faecal zonulin/calprotectin concentrations were measured at 1, 6, 12, 24 months as gut-permeability markers. Zonulin concentrations increased between 1 and 12 months (medians: 114.41, 223.7 ng/mL; respectively), whereas calprotectin concentrations decreased between one and six months (medians: 149. 29, 109.28 µg/mL); both then stabilized (24 months: 256.9 ng/mL zonulin; 59.5 µg/mL calprotectin). In individual children, high levels at one month gave high levels at older ages (correlations: calprotectin: between 1 and 6 or 12 months: correlation coefficient (R) = 0.33, statistical significance (p) = 0.0095; R = 0.28, p = 0.032; zonulin: between 1 and 24 months: R = 0.32; p = 0.022, respectively). Parameters which gave marker increases: antibiotics during pregnancy (calprotectin; six months: by 80%, p = 0.038; 12 months: by 48%, p = 0.028); vaginal birth (calprotectin: 6 months: by 140%, p = 0.005); and > 5.7 pregnancy-BMI increase (zonulin: 12 months: by 74%, p = 0.049). Conclusions: “Closure of the intestines” is spread over time and begins between the sixth and twelfth month of life. Antibiotic therapy, BMI increase > 5.7 during pregnancy and vaginal birth are associated with increased intestinal permeability during the first two years of life. Stool zonulin and calprotectin concentrations were much higher compared with previous measurements at older ages; clinical interpretation and validation are needed (no health associations found).